The latter, i.e., item 4 of Lemma REF , implies an arbitrary function can be expressible as a linear combination of kernel functions in a “reproducing Hilbert space" [1]}.
We should mention that the first above-mentioned property leads to define a displacement-type coherent states [2]}, for generalized annihilation and creation operators. Moreover a Gazeau-Klauder coherent state is defined by the second property and fulfils the third property [2]}. It should be noted that while the latter, i.e., resolution of the identity, fulfils all types of coherent states.
